20201-CTI212 -KALKILUS 2 - 13 (SURYANI) ***

Summary

In this lecture, Professor Suryani discusses Taylor and Maclaurin series in Calculus 2. She explains the definitions, formulas, and processes to derive Taylor series for different functions, such as f(x) = x, f(x) = 1 + x, and f(x) = sin(x). Additionally, she introduces Maclaurin series, which are Taylor series centered at x = 0, and provides examples like f(x) = cos(x) and f(x) = e^x. Detailed explanations and step-by-step calculations are given to help students understand and apply these concepts.

Takeaways

• 📚 The lecture introduces the concept of Taylor and Maclaurin series, which are mathematical tools used to represent functions as an infinite sum of terms calculated from the derivatives of the function at a single point.
• 🔍 The Taylor series is defined for a function 'f(x)' that has derivatives of all orders at a point 'a', and it is represented as a sum of terms involving the derivatives of 'f' evaluated at 'a', each term multiplied by a power of '(x-a)' and divided by the factorial of the power.
• 📘 The Maclaurin series is a special case of the Taylor series where the function is expanded around 'x=0', which simplifies the series since all odd powers of 'x' will have a derivative of zero at 'x=0'.
• 🔢 The script explains the process of finding the Taylor series for a function 'f(x)', starting from finding the first derivative to higher order derivatives, and then substituting these into the Taylor series formula.
• 📝 The factorial function is highlighted as an essential part of the Taylor series formula, where each term involves the factorial of the power of '(x-a)'.
• 📉 The concept of 'remainder' or 'sisa' in the script refers to the difference between the function and its Taylor series approximation, which includes the higher order terms that are not included in the approximation.
• 📌 Specific examples are given to illustrate the process of finding the Taylor series for functions such as 'f(x) = x' at 'x = 2', 'f(x) = sin(x)' at 'x = π/2', and 'f(x) = cos(x)'.
• 📐 The importance of correctly calculating the derivatives and substituting the values of 'a' is emphasized to ensure the accuracy of the Taylor series approximation.
• 📈 The script also discusses the Maclaurin series for functions like 'f(x) = cos(x)', where all derivatives are calculated at 'x = 0' and the series is simplified due to the properties of the cosine function.
• 📚 The lecturer encourages students to engage in discussions with their peers if they have any doubts or do not understand certain concepts, promoting collaborative learning.

Q & A

What is the main topic discussed in the script?

The main topic discussed in the script is Taylor and Maclaurin series, which are mathematical concepts used to approximate functions using their derivatives at a given point.

What is the definition of a Taylor series according to the script?

The Taylor series is defined as an infinite sum of terms calculated from the derivatives of a function evaluated at a single point, divided by the factorial of the order of the derivative, multiplied by the power of the variable.

What is the factorial function mentioned in the script?

The factorial function, denoted as '!', is the product of all positive integers up to a given number, e.g., 5! = 5 × 4 × 3 × 2 × 1.

How is the remainder term in a Taylor series described in the script?

The remainder term in a Taylor series is referred to as the 'suku sisa' in the script, which represents the error term or the difference between the function and its Taylor series approximation.

What is the difference between a Taylor series and a Maclaurin series as explained in the script?

The difference between a Taylor series and a Maclaurin series is that a Maclaurin series is a special case of a Taylor series where the function is expanded around the point x = 0.

What is the process for finding the Taylor series of a function according to the script?

The process involves finding the derivatives of the function up to the desired order, evaluating them at the point 'a', and then substituting these values into the Taylor series formula.

What is the significance of the point 'a' in the Taylor series formula?

The point 'a' is the center around which the function is expanded in the Taylor series. It is the value at which the derivatives are evaluated.

How does the script describe the calculation of the nth derivative of a function?

The script describes the calculation of the nth derivative by repeatedly differentiating the function until the nth order and then simplifying the result.

What is the example given in the script for finding the Taylor series of the function f(x) = sin(x) at x = π/2?

The script provides an example of finding the Taylor series of the sine function at π/2 by calculating its derivatives and substituting the values into the Taylor series formula, taking into account the values of sine and cosine at π/2.

What is the importance of simplifying the factorial terms in the Taylor series calculation as mentioned in the script?

Simplifying the factorial terms is important to reduce the complexity of the calculation and to avoid potential errors in the computation process.

Outlines

00:00

📚 Introduction to Taylor and Maclaurin Series

The first paragraph introduces the concept of Taylor and Maclaurin series in calculus. The speaker, identified as Suryani, a lecturer, begins by explaining the definition of Taylor series, which is a representation of a function as an infinite sum of terms calculated from the derivatives of the function at a single point. The paragraph delves into the mathematical formulation of the series, including the factorial and the sigma notation used in the series expansion. The lecturer also provides an example of finding the Taylor series for the function f(x) = x at x = 2, highlighting the process of calculating the necessary derivatives and substituting them into the series formula.

05:00

🔍 Derivation and Application of Taylor Series

The second paragraph continues the discussion on Taylor series by illustrating the process of deriving and applying the series to different functions. The lecturer explains how to calculate the derivatives of a function and substitute these into the Taylor series formula, taking care to include the factorial terms. An example is given for the function f(x) = sin(x) at x = π/2, where the derivatives of the sine function are calculated up to the sixth derivative. The importance of simplifying the series by substituting the values of the derivatives at the specified point is emphasized, and the concept of the remainder term in the series is briefly touched upon.

10:04

📘 Maclaurin Series and Special Cases

The third paragraph focuses on the Maclaurin series, a special case of the Taylor series where the expansion is centered around x = 0. The lecturer clarifies that the Maclaurin series is simply the Taylor series with all terms evaluated at zero. An example is provided for the cosine function, where the derivatives up to the eighth order are calculated and substituted into the Maclaurin series formula. The paragraph also discusses the simplification of the series by recognizing that odd powers of x will have zero derivatives, thus simplifying the series. The lecturer concludes with a reminder to discuss any unclear points with peers, ending the session with a traditional greeting.

Keywords

💡Taylor Series

The Taylor Series is a mathematical representation of a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point. In the script, the Taylor Series is the central topic, with the lecturer explaining its definition and application to various functions. For example, the script discusses the Taylor Series for the function 'f(x)' at a point 'a', emphasizing its components such as the function's derivatives and the factorial.

💡Derivative

A derivative in calculus represents the rate at which a function changes with respect to its variable. The script mentions finding the first, second, third, and higher-order derivatives of functions to construct their Taylor Series. For instance, when determining the Taylor Series for 'f(x) = x' at 'x = 2', the lecturer calculates the first derivative as '1' and the second derivative as '0', which are then used in the series expansion.

💡Factorial

The factorial of a non-negative integer 'n', denoted by 'n!', is the product of all positive integers less than or equal to 'n'. In the context of the Taylor Series formula, factorials are used to scale the coefficients of the series terms. The script explains that the factorial is integral to the formula, as seen when discussing the coefficients of the series for 'f(x)'.

💡Maclaurin Series

A Maclaurin Series is a special case of a Taylor Series where the function is expanded around the point 'x = 0'. The script distinguishes the Maclaurin Series from the general Taylor Series by setting 'a = 0' and using it to find the series for functions like 'cos(x)'. The讲师 discusses the simplification that occurs when 'x' is substituted with '0', resulting in a series that only includes even powers of 'x'.

💡Function

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The script revolves around functions and their Taylor and Maclaurin series expansions, using specific functions like 'f(x) = x', 'f(x) = sin(x)', and 'f(x) = x^x' to illustrate the concepts.

💡Interval

An interval in mathematics, particularly in calculus, refers to a continuous range of values within which a function is defined or a series converges. The script mentions defining the Taylor Series on a certain interval, indicating the importance of the domain in which the series is valid.

💡Remainder Term

The remainder term in a Taylor Series is the difference between the function and its series approximation. It represents the error introduced by truncating the series to a finite number of terms. The script refers to the 'suku sisa' or remainder when discussing the approximation of functions using the Taylor Series and its truncated terms.

💡Trigonometric Functions

Trigonometric functions are functions of an angle, relating the angles of a triangle to the lengths of its sides. In the script, the讲师uses the sine and cosine functions to demonstrate the Taylor Series, showing how their derivatives can be used to construct their series expansions around 'x = π/2'.

💡Exponential Function

An exponential function is a function of the form 'f(x) = a * b^x', where 'a' and 'b' are constants, and 'b' is positive. The script briefly touches on the exponential function when discussing the Taylor Series, although it does not provide a detailed example.

💡Power Function

A power function is a function of the form 'f(x) = x^n', where 'n' is a real number. The script uses the power function 'f(x) = x^x' to illustrate the process of finding its Taylor Series, highlighting the recursive nature of its derivatives.

Highlights

Introduction to the topic of Taylor and Maclaurin series by Suryani, a lecturer for Calculus 2.

Definition of Taylor series, including its mathematical representation with derivatives up to an infinite order.

Explanation of the factorial function and its role in the Taylor series formula.

The concept of the remainder term in Taylor series and its significance in the approximation of functions.

Derivation of the Taylor series for a function f(x) at a point x=a.

Example calculation of the Taylor series for a function f(x) = x at x = 2.

Step-by-step process of finding derivatives and substituting them into the Taylor series formula.

Importance of considering the factorial in the calculation of each term of the series.

Discussion on the remainder terms and their impact on the accuracy of the series approximation.

Application of the Taylor series to the function f(x) = sin(x) at x = π/2.

Explanation of the periodic nature of trigonometric functions and its effect on their derivatives.

Calculation of the Maclaurin series, a special case of the Taylor series at x = 0.

Derivation of the Maclaurin series for the cosine function, cos(x).

Clarification on the difference between odd and even powers in the Maclaurin series and their derivatives.

Example of deriving the Maclaurin series for a function f(x) = x^x, highlighting the complexity of higher-order derivatives.

Emphasis on the practical applications of Taylor and Maclaurin series in various mathematical and scientific fields.

Encouragement for students to discuss any unclear concepts with their professors for further understanding.

Closing remarks with well-wishes and a reminder of the importance of the material covered.

Transcripts

hai hai

Hai assalamualaikum warahmatullahi

wabarakatuh perkenalkan nama saya

Suryani saya salah satu dosen untuk

matakuliah kalkulus 2 Pada kesempatan

kali ini saya akan membahas mengenai

deret Taylor dan mclaurin definisi dari

deret Taylor itu adalah misalkan Saya

punya suatu fungsi fx gimana FX ini

mempunyai turun ke n artinya dia punya

turunan pertama kuliah turunan kedua

ketiga keempat kelima keenam dan

seterusnya gitu ya sampai dia mempunyai

turunan di suku ke tak hingga dimana

kita mendefinisikan ini pada suatu

interval bukan yang memuat aja dia itu

terdapat di interval buka maka deret

pangkat dari suatu efektif

Hai itu dituliskan seperti ini ya jadi

speak makanya dari nol sampai Infinite

dimana turunan fungsi f purna ke Kak Ya

turun ke Kak dari fungsi f bayi2 bagi

dengan K factory Al seperti itu bacanya

nah kemudian akan faktorial itu

faktorial itu rumusnya adalah qadhi kali

kami satu dikalikan mint dua kali kami 3

dan seterusnya sampai dikali tiga kali

dua kali satu nah itu definisi dari Kak

faktorial Ya dari harusnya di kali sama

x-men ap4k ya Jadi ada esmina ^ tagnya

kemudian kalau seandainya sigmanya saya

Uraikan bentuknya akan seperti ini jadi

Eva dikali F aksen afr-1 faktorial

dikali X min a&d tambah turunan kedua

dititik ar2 faktorial dikali X Mira

faktorial begitu

Nia sampai kita melewatkan turunan ke-n

dari fungsi f dititik a dan seterusnya

jadi masih ada titik-titik ke

belakangnya nah biasanya ini maaf aku

saya deret ini kita sebut dengan deret

Taylor yang dibangkitkan oleh fungsi f

dititik x = a gitu ya Nah untuk

suku-suku yang rendah tarik suku-suku

yang tinggi jadi misalkan turunan ke 8 9

10 itu kita sebut dengan suku sisa ya

jadi suku yang adtu runan yaitu sudah

turunan tinggi udah turun pangkat tinggi

itu kita sebut dengan suku sisa ya Nah

contohnya Tentukan deret Taylor dari

fungsi fx = x di X = 2 ya karena tadi

deret Taylor itu ada unsur turunan

pertama turunan kedua ketiga keempat dan

seterusnya maka saya cari terlebih dulu

turun pertamanya

Hai Natuna pertama dari super X itu

adalah ekspresi tukang batu x pangkat

min 1 ya berarti turunnya pertamanya min

1 per x pangkat 2 atau min x pangkat

minus 2 gitu nah saya tool seperti ini

in 1 x pangkat 2 nah kemudian turunan

kedua itu diperoleh dari turunan pertama

yang kalian turunkan lagi gitu Ini kan

bentuk dari min 1 per x kuadrat itu

adalah x pangkat min 2 min x pangkat min

2 nabati turunannya Min kali min 2 2x

pangkat kurangnya satu batu jadinya mym3

nah saya kulit seperti ini dua per x

pangkat 3 nah begitu juga dengan suruh

pangkat tiga itu diperoleh dari turunan

^ 2-nya turun keduanya kalian turunkan

lagi ya kemudian turunan ke 4 nya Kalau

turun keempat itu hati-hati ya di sini

harus ada kurangnya gitu nah ini kalau

ada kurung dengan tidak ada kurung itu

beda makna jadi kalau ada turun kurung

itu berarti dia adalah turunan ke 4 dari

fungsi fx Kalau tidak ada kurungnya

berarti itu adalah x ^ 4x itu hati-hati

ya Nah kemudian karena tadi kita punya

aa nya itu adalah dua maka Kalian cari

f2f aksen dua F12 N2 begitu seterusnya

ya artinya semua es yang ada di sini

kamu ganti sama2 kemudian disitu Ya

seperti ini nah begitu disini saya

selesaikan sampai turunan ke 4 kalian

boleh menghitung ini sampai Ternate 5

keenam ketujuh gitu nah sisanya itu

adalah suku sisanya ya seperti itu Nah

kita tahu tadi deret Taylor itu kan

bentuknya seperti ini ya Nah kalian

ganti semua XD sama2 eh sorry semuanya

ya semuanya sama2 nah kemudian kalian

subtitusikan nilai yang tadi jadi kita

kan tadi udah punya nilai f2f aksen 2R

13 N2 begitu seterusnya kemudian

dihitung dengan hati-hati di sini ada

faktorialnya ada satu faktorial 2

faktorial tiga faktorial Nah ketika

kalian sudah hitung nah ini adalah

hasilnya jadi setengah dikurangi

seperempat x218 dikali X min 2 kuadrat

Iya begitu seterusnya ditambah

titik-titik ini dimana titik-titik ini

adalah suku sisanya ya Jadi nanti

conditional Kalian mau mengerjakannya

sampai turunan keberapa ya Kemudian yang

kedua aturan dari Thailand dari fungsi

fx = 61 plus X nah ini di X = 1 jadi

prosedur ini sama seperti tadi Kalian

cari dulu Land satu prosesnya cari

turunan pertama turunan pertama dari lem

satu + F1 adalah satu persatu + F

kemudian cari turunan kedua dengan cara

kalian turunkan turunan pertamanya Aa

berarti jadinya minus yah minus 1

R1 + X dikuadrat in kalau turunan

ketiganya berarti dua ya dua persatu + x

^ 3 turunan keempatnya berarti min 6 ya

Nah kita dan disini juga saya

mengerjakannya sampai turunan ke 4

kemudian karena tadi hanya satu maka

ganti semua ST sama satu nah gitu kalian

tinggal ganti semua es teh sama satu nah

kemudian kalau sudah dapatkan nilai R1 F

aksen satu f12e 1 dan seterusnya tinggal

dimasukkan ke ekspansi deret Taylor nya

ya seperti ini nah jangan lupa nanti

dihitung ya di sini ada satu faktorial 2

faktorial tiga faktorial kemudian hasil

akhirnya adalah ini Ok saya lanjutkan

contoh yang ketiga itu adalah the ini

masih tetap deret Taylor Ya tentukan

deret Taylor dari fungsi fx = Sin X DX =

phi per 2

Hai Nah berarti kalian harus cari dulu

turunan pertamanya turunan dari steam

itu adalah tokoh turunan kedua berarti

turunan dari cos Min Sin turunan ketiga

berarti turunan dari bensin X tadi

turunan Sin adalah kos ya berarti Min

cos X turunan ke 4 berarti turunan dari

Min cos X turunan dari cos itu adalah

Min Sin X televisinet balik lagi ya ke

sini ya udah berarti Toure kalimatnya

cos X turunan keenamnya Min Sin X

turunan ketujuhnya Min cos X jadi ya Oh

saya sampai 6 ternyata disini kita jadi

kalau kalian mau menyelesaikan sampai

delapan yang enggak ada masalah lihat

dilanjutkan saja karena tadi esnya itu

saya ambil kiper2 diperbaiki 90° ya jadi

sin90 itu adalah satu cos90 itu adalah

nol ya Jadi tinggal di subtitusikan aja

keturunan pertamanya nah seperti

Ndah kalau kalian sudah memperoleh semua

nilainya tinggal kalian masukkan ke

deret trailernya ekspansi nya ya Nah

seperti itu jangan lupa di Sederhanakan

yah nilai dari satu faktorial 2

faktorial tiga faktorial dan seterusnya

Oke selanjutnya adalah tentang deret

maclaurin Nah jadi deret maclaurin itu

adalah deret Taylor tapi dimana dia itu

dibangkitkan oleh fungsi f DX = 0 jadi

esnya itu spesial x = 0 Jadi kalau ST =

nol dia adalah deret maclaurin ya

seperti itu Nah sehingga ekspansi

deretnya itu ya Nah disini tidak ada

ekstenal lagi karena hanya nol hanya ada

f x kuadrat x ^ 3 begitu seterusnya ya

Nah contohnya Tentukan deret maclaurin

dari fungsi fx = cos X Nah berarti sama

persis ketika kalian mengerjakan deret

Taylor jadi cari dulu turun

Hai pertamanya kedua ketiga keempat

begitu seterusnya Griya turunan dari cos

Min Sin Sunandar istirnya adalag Oh

berarti ada minus ditulis ulang jurnal

dari kos adalah mesin ada minusnya

berarti Sin X turunan Sin adalah chord

nah begitu seterusnya nah ini saya

selesaikan sampai turunan ke-8 karena

disini perintahnya Tentukan deret

maclaurin Berarti semua esnya itu harus

kamu ganti sama nol ya ganti semua x = 0

kemudian kalian hitung itu ya Nah

seperti ini baru kalian substitusikan ke

deret maclaurin ya nah begitu ya Nah

tinggal di substitusikan nilai f0f

Arsenal FC absen nolnya ini nilai yang

sudah dicegah anak Andri hanya sisa suku

ke kenapa kenapa ya jadi x ^ 2x makan 4x

^ X ^

Hai Karena untuk yang ^ ganjil

turunannya itu nilainya nol ya kemudian

contoh berikutnya Tentukan direktori

dari fungsi fx = x ^ x gitu ya Sama

persis Kalian cari turunan pertama kedua

ketiga dan seterusnya kebetulan turun

naik pangkat x itu adalah air pangkat x

turunkan lagi Eh pangkat x lagi gitu ya

ingat karena dia directory kalian ganti

semua X sama ya jadi ketika

disubstitusikan ke deret maclaurin nya

Nah ini adalah hasilnya ya oke demikian

Apa yang dapat saya sampaikan pada

kesempatan kali ini semoga bermanfaat

untuk kalian semua Jika masih ada

hal-hal yang belum kalian pahami silakan

kalian diskusikan dengan dokter

masing-masing ya Sekian dari saya semoga

salamualaikum warahmatullahi wabarakatuh

hai hai